Integrand size = 20, antiderivative size = 98 \[ \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx=-\frac {a B-(A b-a C) x}{4 a b \left (a+b x^2\right )^2}+\frac {(3 A b+a C) x}{8 a^2 b \left (a+b x^2\right )}+\frac {(3 A b+a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}} \] Output:
-1/4*(B*a-(A*b-C*a)*x)/a/b/(b*x^2+a)^2+1/8*(3*A*b+C*a)*x/a^2/b/(b*x^2+a)+1 /8*(3*A*b+C*a)*arctan(b^(1/2)*x/a^(1/2))/a^(5/2)/b^(3/2)
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx=\frac {3 A b^2 x^3-a^2 (2 B+C x)+a b x \left (5 A+C x^2\right )}{8 a^2 b \left (a+b x^2\right )^2}+\frac {(3 A b+a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}} \] Input:
Integrate[(A + B*x + C*x^2)/(a + b*x^2)^3,x]
Output:
(3*A*b^2*x^3 - a^2*(2*B + C*x) + a*b*x*(5*A + C*x^2))/(8*a^2*b*(a + b*x^2) ^2) + ((3*A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2))
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2345, 25, 27, 215, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle -\frac {\int -\frac {3 A b+a C}{b \left (b x^2+a\right )^2}dx}{4 a}-\frac {a B-x (A b-a C)}{4 a b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {3 A b+a C}{b \left (b x^2+a\right )^2}dx}{4 a}-\frac {a B-x (A b-a C)}{4 a b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(a C+3 A b) \int \frac {1}{\left (b x^2+a\right )^2}dx}{4 a b}-\frac {a B-x (A b-a C)}{4 a b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {(a C+3 A b) \left (\frac {\int \frac {1}{b x^2+a}dx}{2 a}+\frac {x}{2 a \left (a+b x^2\right )}\right )}{4 a b}-\frac {a B-x (A b-a C)}{4 a b \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(a C+3 A b) \left (\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b x^2\right )}\right )}{4 a b}-\frac {a B-x (A b-a C)}{4 a b \left (a+b x^2\right )^2}\) |
Input:
Int[(A + B*x + C*x^2)/(a + b*x^2)^3,x]
Output:
-1/4*(a*B - (A*b - a*C)*x)/(a*b*(a + b*x^2)^2) + ((3*A*b + a*C)*(x/(2*a*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])))/(4*a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.47 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {\frac {\left (3 A b +C a \right ) x^{3}}{8 a^{2}}+\frac {\left (5 A b -C a \right ) x}{8 a b}-\frac {B}{4 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 A b +C a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a^{2} b \sqrt {a b}}\) | \(83\) |
risch | \(\frac {\frac {\left (3 A b +C a \right ) x^{3}}{8 a^{2}}+\frac {\left (5 A b -C a \right ) x}{8 a b}-\frac {B}{4 b}}{\left (b \,x^{2}+a \right )^{2}}-\frac {3 A \ln \left (b x +\sqrt {-a b}\right )}{16 \sqrt {-a b}\, a^{2}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) C}{16 \sqrt {-a b}\, b a}+\frac {3 A \ln \left (-b x +\sqrt {-a b}\right )}{16 \sqrt {-a b}\, a^{2}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) C}{16 \sqrt {-a b}\, b a}\) | \(153\) |
Input:
int((C*x^2+B*x+A)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
(1/8*(3*A*b+C*a)/a^2*x^3+1/8*(5*A*b-C*a)/a/b*x-1/4*B/b)/(b*x^2+a)^2+1/8*(3 *A*b+C*a)/a^2/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
Time = 0.08 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.20 \[ \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx=\left [-\frac {4 \, B a^{3} b - 2 \, {\left (C a^{2} b^{2} + 3 \, A a b^{3}\right )} x^{3} + {\left ({\left (C a b^{2} + 3 \, A b^{3}\right )} x^{4} + C a^{3} + 3 \, A a^{2} b + 2 \, {\left (C a^{2} b + 3 \, A a b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (C a^{3} b - 5 \, A a^{2} b^{2}\right )} x}{16 \, {\left (a^{3} b^{4} x^{4} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2}\right )}}, -\frac {2 \, B a^{3} b - {\left (C a^{2} b^{2} + 3 \, A a b^{3}\right )} x^{3} - {\left ({\left (C a b^{2} + 3 \, A b^{3}\right )} x^{4} + C a^{3} + 3 \, A a^{2} b + 2 \, {\left (C a^{2} b + 3 \, A a b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (C a^{3} b - 5 \, A a^{2} b^{2}\right )} x}{8 \, {\left (a^{3} b^{4} x^{4} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2}\right )}}\right ] \] Input:
integrate((C*x^2+B*x+A)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
[-1/16*(4*B*a^3*b - 2*(C*a^2*b^2 + 3*A*a*b^3)*x^3 + ((C*a*b^2 + 3*A*b^3)*x ^4 + C*a^3 + 3*A*a^2*b + 2*(C*a^2*b + 3*A*a*b^2)*x^2)*sqrt(-a*b)*log((b*x^ 2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*(C*a^3*b - 5*A*a^2*b^2)*x)/(a^3*b ^4*x^4 + 2*a^4*b^3*x^2 + a^5*b^2), -1/8*(2*B*a^3*b - (C*a^2*b^2 + 3*A*a*b^ 3)*x^3 - ((C*a*b^2 + 3*A*b^3)*x^4 + C*a^3 + 3*A*a^2*b + 2*(C*a^2*b + 3*A*a *b^2)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + (C*a^3*b - 5*A*a^2*b^2)*x)/(a ^3*b^4*x^4 + 2*a^4*b^3*x^2 + a^5*b^2)]
Time = 0.55 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.59 \[ \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx=- \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \cdot \left (3 A b + C a\right ) \log {\left (- a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \cdot \left (3 A b + C a\right ) \log {\left (a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} + x \right )}}{16} + \frac {- 2 B a^{2} + x^{3} \cdot \left (3 A b^{2} + C a b\right ) + x \left (5 A a b - C a^{2}\right )}{8 a^{4} b + 16 a^{3} b^{2} x^{2} + 8 a^{2} b^{3} x^{4}} \] Input:
integrate((C*x**2+B*x+A)/(b*x**2+a)**3,x)
Output:
-sqrt(-1/(a**5*b**3))*(3*A*b + C*a)*log(-a**3*b*sqrt(-1/(a**5*b**3)) + x)/ 16 + sqrt(-1/(a**5*b**3))*(3*A*b + C*a)*log(a**3*b*sqrt(-1/(a**5*b**3)) + x)/16 + (-2*B*a**2 + x**3*(3*A*b**2 + C*a*b) + x*(5*A*a*b - C*a**2))/(8*a* *4*b + 16*a**3*b**2*x**2 + 8*a**2*b**3*x**4)
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (C a b + 3 \, A b^{2}\right )} x^{3} - 2 \, B a^{2} - {\left (C a^{2} - 5 \, A a b\right )} x}{8 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} + \frac {{\left (C a + 3 \, A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b} \] Input:
integrate((C*x^2+B*x+A)/(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/8*((C*a*b + 3*A*b^2)*x^3 - 2*B*a^2 - (C*a^2 - 5*A*a*b)*x)/(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b) + 1/8*(C*a + 3*A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a* b)*a^2*b)
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (C a + 3 \, A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b} + \frac {C a b x^{3} + 3 \, A b^{2} x^{3} - C a^{2} x + 5 \, A a b x - 2 \, B a^{2}}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b} \] Input:
integrate((C*x^2+B*x+A)/(b*x^2+a)^3,x, algorithm="giac")
Output:
1/8*(C*a + 3*A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b) + 1/8*(C*a*b*x^3 + 3*A*b^2*x^3 - C*a^2*x + 5*A*a*b*x - 2*B*a^2)/((b*x^2 + a)^2*a^2*b)
Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {x^3\,\left (3\,A\,b+C\,a\right )}{8\,a^2}-\frac {B}{4\,b}+\frac {x\,\left (5\,A\,b-C\,a\right )}{8\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (3\,A\,b+C\,a\right )}{8\,a^{5/2}\,b^{3/2}} \] Input:
int((A + B*x + C*x^2)/(a + b*x^2)^3,x)
Output:
((x^3*(3*A*b + C*a))/(8*a^2) - B/(4*b) + (x*(5*A*b - C*a))/(8*a*b))/(a^2 + b^2*x^4 + 2*a*b*x^2) + (atan((b^(1/2)*x)/a^(1/2))*(3*A*b + C*a))/(8*a^(5/ 2)*b^(3/2))
Time = 0.16 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.16 \[ \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^3} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} c +6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} x^{2}+2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c \,x^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} x^{4}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} c \,x^{4}+5 a^{2} b^{2} x -2 a^{2} b^{2}-a^{2} b c x +3 a \,b^{3} x^{3}+a \,b^{2} c \,x^{3}}{8 a^{2} b^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((C*x^2+B*x+A)/(b*x^2+a)^3,x)
Output:
(3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b + sqrt(b)*sqrt(a)* atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*c + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt( b)*sqrt(a)))*a*b**2*x**2 + 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a))) *a*b*c*x**2 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*x**4 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**2*c*x**4 + 5*a**2*b**2*x - 2*a**2*b**2 - a**2*b*c*x + 3*a*b**3*x**3 + a*b**2*c*x**3)/(8*a**2*b**2*( a**2 + 2*a*b*x**2 + b**2*x**4))